@Article{CSIAM-AM-4-275,
author = {Lin , GuochangHu , PipiChen , FukaiChen , XiangChen , JunqingWang , Jun and Shi , Zuoqiang},
title = {BINet: Learn to Solve Partial Differential Equations with Boundary Integral Networks},
journal = {CSIAM Transactions on Applied Mathematics},
year = {2023},
volume = {4},
number = {2},
pages = {275--305},
abstract = {<p style="text-align: justify;">We propose a method combining boundary integral equations and neural
networks (BINet) to solve (parametric) partial differential equations (PDEs) and operator problems in both bounded and unbounded domains. For PDEs with explicit
fundamental solutions, BINet learns to solve, as a proxy, associated boundary integral equations using neural networks. The benefits are three-fold. Firstly, only the
boundary conditions need to be fitted since the PDE can be automatically satisfied
with single or double layer potential according to the potential theory. Secondly, the
dimension of the boundary integral equations is less by one, and as such, the sample
complexity can be reduced significantly. Lastly, in the proposed method, all differential operators have been removed, hence the numerical efficiency and stability are
improved. Adopting neural tangent kernel (NTK) techniques, we provide proof of
the convergence of BINets in the limit that the width of the neural network goes to
infinity. Extensive numerical experiments show that, without calculating high-order
derivatives, BINet is much easier to train and usually gives more accurate solutions,
especially in the cases that the boundary conditions are not smooth enough. Further,
BINet outperforms strong baselines for both one single PDE and parameterized PDEs
in the bounded and unbounded domains.</p>},
issn = {2708-0579},
doi = {https://doi.org/10.4208/csiam-am.SO-2022-0014},
url = {http://global-sci.org/intro/article_detail/csiam-am/21415.html}
}